Timeline for PNT for general zeta functions, Applications of.
Current License: CC BY-SA 2.5
8 events
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| Nov 27, 2018 at 9:50 | comment | added | reuns | To not use Artin L-functions explicitly, I guess Chebotarev was proving the PNT for $\prod_p (1+f(h_p)p^{-s})$ where $h_p$ is the polynomial such that $\zeta_p(K,s) = h_p(p^{-s})^{-1}$ ? This way he missed that for a basis of such $f$ (the Artin L-functions) there was a meromorphic continuation and a functional equation (as from Brauer theorem and class field theory they are quotients of products of Hecke L-functions) @KConrad | |
| Feb 6, 2010 at 2:22 | comment | added | KConrad | To add another detail to Matt's clarification, even the Riemann zeta-function tells you something from its pole at s = 1: there are infinitely many primes. If you want prime asymptotics (in the sense of natural density) then you have to deal with behavior on the whole line Re(s) = 1, and likewise for Dirichlet's theorem in its qualitative (or analytic density) form vs. in its natural density form. | |
| Feb 5, 2010 at 18:42 | comment | added | Emerton | I mean the versions of Dirichlet and Cebotarev with the correct asymptotics. As for Sato--Tate, an examination of the discussion in Serre's book will answer your question. | |
| Feb 5, 2010 at 18:40 | history | edited | Emerton | CC BY-SA 2.5 |
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| Feb 5, 2010 at 18:08 | comment | added | Anweshi | Dear prof. Emerton., Your answer was very helpful nevertheless, and I have upvoted it. | |
| Feb 5, 2010 at 18:05 | comment | added | Anweshi | Deaer Emerton. Dirichlet's theorem on arithmetic progression follows just from the fact that the Dedekind zeta function has a pole at $s = 1$. It is not an analogue of PNT; one does not have to do any of contour integration I described, or difficult and hard estimates. It is a much simpler theorem in analytic number theory. Chebotarev is just the generalization of Dirichlet's theorem, not of PNT. The Sato-Tate is looking more closely at the error term in the Weil bound, how is this a generalization of PNT?? | |
| Feb 5, 2010 at 17:13 | history | edited | Emerton | CC BY-SA 2.5 |
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| Feb 5, 2010 at 15:26 | history | answered | Emerton | CC BY-SA 2.5 |